Bessel potential

Summary

The Bessel potential of order $\alpha > 0$ is the operator $(I - \Delta)^{-\alpha/2}$ on functions on $\R^n$, where $\Delta$ is the Laplacian. It is a convolution with a positive integrable kernel $G_\alpha$. Near the origin $G_\alpha$ behaves like the kernel of the Riesz potential. It decays exponentially at infinity, which removes the problems of the Riesz potential at large scales. The images of $L^p$ under Bessel potentials are the Bessel potential spaces, which give Sobolev spaces of fractional order.